2011-2012 Four functions are continuous and differentiable for all real numbers, and some of their values (and the values of their derivatives) are presented in the below table: If you know that h(x) = f(x) · g(x) and j(x) = g(f(x)), fill in the correct numbers for each blank value in the table. Solution: Woo, … Continue reading "Problem 7: Revenge of Table Derivatives"
2011-2012 Particle Motion Definition and Calculus The position of a particle (in inches) moving along the x-axis after t seconds have elapsed is given by the following equation: s = f(t) = t4 – 2t3 – 6t2 + 9t (a) Calculate the velocity of the particle at time t. (b) Compute the particle’s velocity at t … Continue reading "Problem 5: Particle Motion Definition and Calculus"
2011-2012 During a taping for Circus of the Stars, beloved actress Betty White is shot out of a cannon. The firing goes completely awry and sends her on a collision course with a jet. As they converge, Betty and the jet plane at right angles to each other (see diagram below). Betty is 200 miles … Continue reading "Problem 4: B-B-B-Betty and the Jet (Related Rates)"
2011-2012 Let f(x) be the function defined below: Determine whether f(x) is continuous at x = 0 and explain your answer. Note: You may use a graphing calculator to examine the graph of f(x). Solution: If f(x) if continuous at x = 0, its left- and right-hand limits exist at x = 0, and they … Continue reading "Problem 2: Continuity"
2011-2012 Describe or draw a function, f(x), with the following characteristics: f(x) has domain (–∞,8) f(x) has range (–∞,9) f(4) = 0; f(5) = 0; f(7) = 0 The limit, as x approaches –∞, of f(x) equals 9 The limit, as x approaches 8 from the left, of f(x) equals –∞ f(–1) = f(–3); f(–1) … Continue reading "Problem 1: Limits"